Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
CfE Advanced Higher Physics – Unit 3 – Electromagnetism
FIELDS
1. Electric field strength
2. Coulomb’s inverse square law
3. Electric potential and electric field strength around a point charge and a system of
charges
4. Potential difference and electric field strength for a uniform field
5. Motion of charged particles in uniform electric fields
6. The electronvolt as a unit of energy
7. Ferromagnetism
8. Magnetic field patterns
9. Magnetic induction
10. Magnetic induction at a distance from a long current carrying wire
11. Force on a current carrying conductor in a magnetic field
12. Compare gravitational, electrostatic, magnetic and nuclear forces
CIRCUITS
13. Capacitors in d.c. circuits
14. Time constant for a CR circuit
15. Capacitors in a.c. circuits
16. Capacitive reactance
17. Inductors in d.c. circuits
18. Self-inductance of a coil
19. Lenz’s law
20. Energy stored by an inductor
21. Inductors in a.c. circuits
22. Inductive reactance
ELECTROMAGNETIC RADIATION
23. Knowledge of the unification of electricity and magnetism
24. Electromagnetic radiation exhibits wave properties
25. Electric and magnetic field components of electromagnetic radiation
26. Relationship between the speed of light and the permittivity and permeability of free
space.
1
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
ELECTRIC FIELDS
Forces between Electric Charges - Coulomb's Law (1785)
Forces between electric charges have been observed since earliest times. Thales of
Miletus, a Greek living in around 600 B.C., observed that when a piece of amber was
rubbed, the amber attracted bits of straw. It was not until 2500 years later, however,
that the forces between charged particles were actually measured by Coulomb using a
torsion balance method. The details of Coulomb's experiment are interesting but his
method is difficult to reproduce in a teaching laboratory.
Coulomb's Inverse Square Law
Coulomb's experiment gives the following mathematical results:
1
F∝ r2
F∝Q1 ×Q2
and
Thus
F k
Q1 Q2
r2
Where r is the separation between two charges, Q1 and Q2.
Value of k
When other equations are developed from Coulomb's Law, it is found that the product
4πk frequently occurs. Thus, to avoid having to write the factor 4π in these derived
equations, it is convenient to define a new constant , called the permittivity of free
space and is equal to 8.85 x 10-12 F m-1, such that:
where ε is the Greek letter 'epsilon'
or
k is approximately 9.0 x 109 N m2 C-2
Equation for Coulomb's inverse square law
when Q1 and Q2 are separated by air or a vacuum
Notes:
• Force is a vector quantity. If more than two charges are present, the force on any
given charge is the vector sum of all the forces acting on that charge.
• Coulomb's law has a similar form as the gravitational force,
2
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Example
Three identical charges A, B and C are fixed at the positions shown in the right
angled triangle below.
Each charge is +8 nC (i.e. +8.0 x 10-9 C) in magnitude.
(a)
Calculate the forces exerted on charge A by charges B and C.
(b)
Calculate the resultant force on charge A. (This means magnitude and
direction)
Solution
(a)
FBA 1 Q1 Q2
4πε0 r 2
9×109 ×
8×10-9 ×8×10-9
$0.6'2
1.6×10-6 N
Direction is along BA, giving repulsion.
1 Q1 Q2
FCA 4πε
0
r2
9×109 ×
8×10-9 ×8×10-9
$0.8'2
0.9×10-6 N
Direction is along CA, giving repulsion
Resultant 2FBA 2 3FCA 2 1.8×10-6 N
0.9×10-4
tanθ 1.6×10-4 0.563
Thus θ = 29° (this is not a bearing)
Resultant force on charge A = 1.8 x 10-6 N at an angle of 29° as shown above.
The table below contains some atomic data for answering Coulombs Law questions.
Typical
Typical
diameter of diameter of
atoms (m) nuclei (m)
Particle
Symbol
Charge (C)
Mass (kg)
proton
p
+e
1.60 x 10-19
1.673 x 10-27
1 x 10-10
1 x 10-15
neutron
n
0
1.675 x 10-27
to
To
electron
e-
-e
-1.60 x 10-19
9.11 x 10-31
3 x 10-10
7 x 10-15
3
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
The Electric Field
The idea of a field is used to describe or visualise how objects at a distance affect one
another. In terms of electric fields we say that a charge sets up a field around itself such
that it will influence other charges present in that field.
Charge Qt placed at point P in the field caused by charge Q, will experience a force F
due to the presence and strength of the field at P. A charged object does not experience
its own electric field.
Definition of an electric field
An electric field is said to be present at a point or location if a force, of electrical origin,
is exerted on a charge placed at that point.
In the following work on electric fields there are two parts to the problem:
• calculating the fields set up by certain charge distributions
• calculating the force experienced by a charge when placed in a known field.
Electric Field Strength
The electric field strength E at any point is the force experience by a unit positive
charge placed at that point.
If a charge Qt, placed at point P in the electric field, experiences a force F then:
F
E Q
t
E has units of N C-1
• The direction of E is conventionally taken as the direction in which a positive test
charge would move in the field. Thus in the presence of a positive charge, the
direction of the field is away from that charge, and vice versa.
• The charge Qt must be small enough not to alter the field, E.
• The unit N C-1 is equivalent to the unit V m-1, see later.
F
This is similar to a gravitational field around mass: g m where g has the unit N kg-1.
Electric Field Lines
An electric field line is a convenient concept developed by Michael Faraday to help the
visualisation of an electric field.
• The tangent to a field line at a point gives the direction of the field at that point.
• Field lines are continuous; they begin on positive charges and end on negative
charges. They cannot cross.
• If consecutive field lines are close together then the electric field strength is strong,
if the lines are far apart the field is weak. If lines are parallel and equally spaced the
field is said to be uniform.
• Field lines cut equipotential surfaces at right angles, see later.
4
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Examples of Electric Field Patterns
isolated positive charge
isolated negative charge
These patterns are called radial fields. The lines are like the radii of a circle.
Two equal but opposite charges
Charged parallel plates
The field lines are parallel and
equally spaced between the plates.
This is called a uniform field.
Equation for Electric Field Strength
Consider placing test charge Qt at a point distance r from a fixed point charge Q in a
vacuum.
The force between the two charges is given by: F
Force
Electric field strength, E is defined as Charge thus E
E
1 QQt
4πε0 r 2
F
Qt
giving:
1 Q
4πε0 r 2
• This equation gives the magnitude of the electric field strength around an isolated
point charge; its direction is radial. The electric field strength reduces quickly as the
1
distance, r, increases because E∝ r2.
• Electric field strength is a vector quantity. When more than one charge is present,
the electric field strength must be calculated for each charge and the vector sum of
their effects influencing the field at that point determined.
5
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Example: The Electric Dipole
A pair of charges +4.0 x 10-9 C and -4.0 x 10-9 C separated by 2.0 x 10-14 m make up
an electric dipole. Calculate the electric field strength at the point P, a distance of
5.0 x 10-14 m from the dipole along the axis shown.
Solution
1
In magnitude E1 E2 4πε
0
Q
r2
1
4πε
Horizontally: E1 sin θ + E2 sin θ = 0
Vertically:
0
4.0×10-9
$5.1×10-14 '2
1.38×1028 NC-1
since E1 and E2 are in opposite directions.
= >
. ×
EP = 2 E1 cos θ = 2 x 1.38 x 1028 x ?.
Thus EP = 5.4 x 1027 N C-1
= >
×
5.4 × 10@A
The direction of EP is given in the sketch above.
A knowledge of electric dipoles is important when trying to understand the behaviour
of dielectric materials which are used in the construction of capacitors.
An analysis of the water molecule also shows that there is a resultant electric field
associated with the oxygen atom and two hydrogen atoms - water is known as a polar
molecule.
Potential Difference and Electric Field Strength for a uniform field
For a uniform field the electric field strength is the same at all points.
The potential difference between two points is the work done
in moving one coulomb of charge from one point to the other
against the electric field, i.e. from the lower plate to the
upper plate. The minimum force needed to move Q
coulombs from the lower plate to the upper plate is QE.
Thus
work = force x distance = QE x d. but work = QV by definition
thus QV = QE x d
V=Ed
for a uniform field.
V
An alternative equation for electric field strength is E d with a unit of V m-1,
showing that the unit N C-1 is equivalent to the unit V m-1 as mentioned earlier.
6
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Conducting Shapes in Electric Fields
As shown in the Faraday's Ice-Pail Experiment any charge given to a conductor always
resides on the outer surface of the conductor. A direct consequence of this fact is that
the electric field inside a conductor must be zero, that is Einside = 0.
Reasoning
The field must be zero inside the conductor because if it were non-zero any charges
placed inside would accelerate in the field and move until balance was reached again.
This would only be achieved when no net force acted on any of the charges, which in
turn means that the field must be zero. This is also why any excess charge must reside
entirely on the outside of the conductor with no net charge on the inside. The field
outside the conductor must start perpendicular to the surface. If it did not there would
be a component of the field along the surface causing charges to move until balance
was reached. If an uncharged conductor is placed in an electric field, charges are
induced as shown below so that the internal field is once again zero. Notice that the
external field is modified by the induced charges on the surface of the conductor and
that the overall charge on the conductor is still zero.
You can now see why the leaf deflection of a charged gold leaf electroscope can go
down if an uncharged metal object is brought close - the field set up by the charge on
the electroscope causes equal and opposite charges to be induced on the object.
Electrostatic shielding
If the conductor is hollow then the outer surface acts as a "screen" against any external
electric field. This principle is used in co-axial cables (shown below).
The 'live' lead carrying the signal is shielded from external electric fields, i.e.
interference, by the screen lead which is at zero volts.
7
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Electrostatic Potential
To help understand this concept consider the sketch below.
To move Qt from a to b requires work from an external agent, e.g. the moving belt of a
Van de Graaff machine. The work supplied increases the electrostatic potential
energy of the system. This increase of energy depends on the size of the charge Q and
on the positions a and b in the field.
Definition of electrostatic potential
Let an external agent do work, W, to bring a positive test charge, Qt, from infinity to a
point in an electric field.
The electrostatic potential, V, is defined to be the work done by external forces in
bringing unit positive charge from infinity to that point.
Thus
W
V Q
t
the units of electrostatic potential are J C-1
W Qt V
A potential exists at a point a distance r from a point charge; but for the system to have
energy, a charge must reside at the point. Thus one isolated charge has no electrostatic
potential energy.
8
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Electrostatic Potential due to a Point Charge
To find the electrostatic potential at a point, P, a distance, r, from the charge, Q, we
need to consider the work done to bring a small test charge Qt from infinity to that point.
The force acting against the charge Qt increases as it comes closer to Q. Calculus is
used to derive the following expression for the electrostatic potential V at a distance r
due to a point charge Q.
1 Q
V 4πε
0
r
thus
1
V∝ r
Notice that the expression for electrostatic potential has a very similar form to that for
m
gravitational potential: V -G r .
• Electrostatic potential is a scalar quantity. If a number of charges lie close to one
another the potential at a given point is the scalar sum of all the potentials at that
point. This is unlike the situation with electric field strength. Negative charges have
a negative potential.
• In places where E = 0, V must be a constant at these points. We will see this later
when we consider the field and potential around charged spheres.
Electrostatic Potential Energy
Electrostatic potential at P is given by
V Q
4πε0 F
If a charge Qt is placed at P
Q
electrostatic potential energy of charge Qt
:
E Qt × 4πε
electrostatic potential energy of charge Qt
:
E 4πεt
QQ
0r
0r
A positively charged particle, if free to move in an electric field, will accelerate in the
direction of the field lines. This means that the charge is moving from a position of
high electrostatic potential energy to a position of lower electrostatic potential energy,
losing electrostatic potential energy as it gains kinetic energy.
The Electronvolt
This is an important unit of energy in high energy particle physics.
The electronvolt is the energy acquired when one electron accelerates through a
potential difference of 1 V. This energy, QV, is changed from electrical to kinetic
energy.
1 electronvolt = 1.6 x 10-19 C x 1 V giving
1 eV = 1.6 x 10-19 J.
Often the unit MeV is used;
1 MeV = 1.6 x 10-13 J.
9
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Equipotentials
This idea of potential gives us another way of describing fields. The first approach was
to get values of E, work out the force F on a charge and draw field lines. A second
approach is to get values of V, work out the electrostatic potential at a point and draw
equipotential lines or surfaces.
Equipotential surfaces are surfaces on which the potential is the same at all points; that
is no work is done when moving a test charge between two points on the surface. This
being the case, equipotential surfaces and field lines are at right angles.
The sketches below show the equipotential surfaces (broken lines) and field lines (solid
lines) for different charge distributions. These diagrams show 2-dimensional pictures
of the field. The field is of course 3-dimensional.
(a) isolated positive charge
(b) isolated negative charge
(c) two unlike charges
(d) two like charges
10
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Charged Spheres
For a hollow or solid sphere any excess charge will be found on its outer surface.
The following graphs show the variation of both electric field strength and electrostatic
potential with distance for a sphere carrying an excess of positive charges.
The main points to remember are:
• the electric field is zero inside the sphere
• outside the sphere the electric field varies as the inverse square of distance from
1
sphere; E∝ 2 .
r
• the potential has a constant (non-zero) value inside the sphere
• Outside the sphere the potential varies as the inverse of the distance from the sphere;
1
V∝ .
r
Graphs of Electric Field and Electrostatic Potential
If a sphere, of radius a, carries a charge of Q coulombs the following conditions apply:
Q
Eoutside 4πε
(where r > a),
2
0r
1 Q
Voutside 4πε
0
r
Q
Esurface 4πε
(where r > a),
0a
2
,
and
Einside = 0.
1
Vsurface Vinside 4πε
0
Q
a
Applications of Electrostatic Effects
There are a number of devices which use electrostatic effects, for example, copying
machines, laser printers, electrostatic air cleaners, lightning conductors and electrostatic
generators.
11
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Movement of Charged Particles in Uniform Electric Fields
Charge moving perpendicular to the plates
The particle, mass m and charge Q, shown in the
diagram opposite will experience an acceleration
upwards due to an unbalanced electrostatic force.
Here the weight is negligible compared to the
electrostatic force. The particle is initially at rest.
F
EQ
a (acceleration uniform because E
m
m
uniform)
E is only uniform if length l >> separation d
Ek acquired by the particle in moving distance d = Work done by the electric force
change in Ek = F x displacement
1
mv2 - 0
2
giving the speed at the top plate, v 2
2QEd
m
= Fxd
where F = EQ
notice that Ed also = V
Alternatively the equation for a charged particle moving through voltage V is
1
2
mv 2 QV
and
v 2
2QV
m
Charge moving parallel to the plates
Consider an electron, with initial speed u, entering a uniform electric field mid-way
between the plates:
1
using s ut3 2 at 2
horizontally: x = ut (no force in x direction)
1
vertically: y at 2 (uy = 0 in y direction)
2
1
x2
substituting t: y 2 a u2
F
Now, a m Thus
QE
m
eE
m
eE
y L2mu2 M .x 2
Now since e, E, m and u are all constants we can say:
y = (constant) . x2
This is the equation of a parabola. Thus the path of an electron passing between the
parallel plates is a parabolic one, while the electron is between the plates. After it leaves
the region of the plates the path of the electron will be a straight line.
(Note: there is no need to remember this formula. You can work out solutions to
problems from the basic equations. This type of problem is similar to projectile
problems).
Applications of electrostatic deflections, in addition to those mentioned previously:
e
•
deflection experiments to measure charge to mass ratio for the electron; m
•
e
the cathode ray oscilloscope.
12
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Example - The Ink-Jet Deflection
The figure below shows the deflecting plates of an ink-jet printer. (Assume the ink drop
to be very small such that gravitational forces may be neglected).
An ink drop of mass 1.3 x 10-10 kg, carrying a charge of 1.5 x 10-13 C enters a deflecting
plate system with a speed u = 18 m s-1. The length of the plates is 1.6 x 10-2 m and the
electric field between the plates is 1.4 x 106 N C-1. Calculate the vertical deflection, y,
of the drop at the far edge of the plates.
Solution
a t $1.5×10-13 '×$1.4×106 '
Fe
QE
1615ms -2
m
m
1.3×10-10
x
1.6×10-2
8.9×10-4 s
ux
18
1
1
y at 2 ×1615×$8.9×10-4 '2 6.4×10-4 m 6.4mm
2
2
This method can be used for the deflection of an electron beam in a cathode ray tube.
Relativistic Electrons
You may notice that the velocity of an electron which accelerates through a potential
difference of 1 x 106 V works out to be 6.0 x 108 m s-1. This is twice the speed of light!
The equation used sets no limits on the speed of a charged particle - this is called a
classical equation. The correct equation requires us to take Special Relativity effects
into account.
The equation E = mc2 applies equally well to stationary and moving particles.
Consider a particle, charge Q, accelerated through a potential difference V.
It is given an amount of kinetic energy QV in addition to its rest mass energy (moc2),
so that its new total energy (E) is moc2 + QV.
General relativistic equation
The general equation for the motion of a charged particle is given below.
m
mc2 = moc2 + QV
where m =
o
2
1-
v
2
c
If we take the voltage quoted above, 1 x 106 V, and use the relativistic equation we find
that the speed of the electron works out to be 2.82 x 108 m s-1 or v = 0.94c. You should
check this for yourself.
Relativistic effects must be considered when the velocity of a charged particle is more
than 10% of the velocity of light.
13
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Head-On Collision of Charged Particle with a Nucleus
In the situation where a particle with speed v and positive charge q has a path which
would cause a head-on collision with a nucleus of charge Q, the particle may be brought
to rest before it actually strikes the nucleus. If we consider the energy changes involved
we can estimate the distance of closest approach of the charged particle. At closest
approach change in Ek of particle = change in Ep of particle.
Position
Kinetic energy
Electrostatic
potential energy
1
mv 2
2
infinity
closest
approach
0
qQ 1
4peo r
0
1
1
Change in
EK 2 mv 2 -0 2 mv 2
Change in electrostatic
EP 4πε
qQ 1
0
r
qQ 1
-0 4πε
0
r
Change in Ek = change in electrostatic Ep
qQ 1
4πε0 r
1
mv 2
2
2qQ
r 4πε
and rearranging
0 mv
2
Example
Fast moving protons strike a glass screen with a speed of 2.0 x 106 m s-1. Glass is largely
composed of silicon which has an atomic number of 14. Calculate the closest distance
of approach that a proton could make in a head-on collision with a silicon nucleus.
Solution
1
Using mv 2
2
qQ 1
4πε0 r
gives
r 2qQ
and
4πε0 mv 2
4πε0
9.0 × 10Q
here q = 1.6 x 10-19 C and Q = 14 x 1.6 x 10-19 C (i.e. equivalent of 14 protons)
r 9.0×109 ×
2×R1.6×10-19 S×14×R1.6×10-19 S
1.97×10-27 ×R2.0×106 S
2
r 9.7×10-13 m
14
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Millikan's Oil-Drop Experiment (1910 - 1913)
If possible, view a simulation of this experiment before reading this note.
The charge on the electron was measured by Millikan in an ingenious experiment. The
method involved accurate measurements on charged oil drops moving between two
charged parallel metal plates as shown below.
Tiny oil drops are charged as they leave the atomiser.
As the droplets fall they quickly reach their terminal velocity and, as a result, a steady
speed. An accurate measurement of this speed allows a value for the radius of the drop
to be calculated. From this radius the volume is found, and using the density of the oil,
the mass of the drop is discovered. The drop can be kept in view by adjusting the
voltage between the plates. For the polarity shown above, negatively charged oil drops,
can be held within the plates.
The second part of each individual experiment involved finding the p.d. needed to
'balance' the oil drop (gravitational force equal and opposite to the electric force).
Therefore
mg QE
and
E
V
d
giving
Q mgd
V
Analysis of results
Millikan and his assistants experimented on thousands of oil drops and when all these
results were plotted it was obvious that all the charges were multiples of a basic charge.
This was assumed to be the charge on the electron. Single electron charges were rarely
observed and the charge was deduced from the gaps between 'clusters' of results where
Q = ne (n = ±1, ±2, ±3 etc.)
Conclusions
• Any charge must be a multiple of the electronic charge, 1.6 x 10-19 C. Thus we say
that charge is 'quantised', that is it comes in quanta or lumps all the same size.
• It is not possible to have a charge of, say, 2.4 x 10-19 C because this would involve a
fraction of the basic charge.
15
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Boroughmuir High School
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Advanced Higher Physics
Electromagnetism
Magnetism
Introduction
Modern electromagnetism as we know it started in 1819 with the discovery by the
Danish scientist Hans Oersted that a current-carrying wire can deflect a compass needle.
Twelve years afterwards, Michael Faraday and Joseph Henry discovered
(independently) that a momentary e.m.f. existed across a circuit when the current in a
nearby circuit was changed. It was also discovered that moving a magnet towards or
away from a coil produced an e.m.f. across the ends of the coil. Thus the work of
Oersted showed that magnetic effects could be produced by moving electric charges
and the work of Faraday and Henry showed that an e.m.f. could be produced by
moving magnets.
All magnetic phenomena arise from forces between electric charges in motion. Since
electrons are in motion around atomic nuclei, we can expect individual atoms of all the
elements to exhibit magnetic effects and in fact this is the case. In some metals like
iron, nickel, cobalt and some rare earths these small contributions from individual
atoms can be made to 'line up' and produce a detectable magnetic property. This
property is known as ferromagnestism.
The Magnetic Field
As you have seen from gravitational and electrostatics work, the concept of a field is
introduced to deal with 'action-at-a-distance' forces.
Permanent Magnets
It is important to revise the field patterns around some of the combinations of bar
magnet. (You can confirm these patterns using magnets and iron filings to show up
the field lines, although not their directions).
isolated bar magnet
opposite poles adjacent
like poles adjacent
To establish which end of a bar magnet is the north (N) pole, float the magnet on cork
or polystyrene in a bowl of water and the end which points geographically north is the
'magnetic north'. Similarly a compass needle, which points correctly towards
geographic north, will point towards the magnetic south pole of a bar magnet. Thus a
compass needle will show the direction of the magnetic field at a point which is defined
to be from magnetic north to south.
Electromagnets
A magnetic field exists around a moving charge in addition to its electric field. A
charged particle moving across a magnetic field will experience a force.
16
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Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Magnetic field patterns
A straight wire
Before the current is switched on
the compass needles will point north.
A coil (solenoid) Earth’s Magnetic Field
Notice the almost uniform
field inside the coil.
Left hand grip rule
The direction of the magnetic field, (the magnetic induction,
see below) around a wire is given by the left hand grip rule as
shown below.
Direction (Left Hand Grip Rule)
Grasp the current carrying wire in your left hand with your
extended left thumb pointing in the direction of the electron
flow in the wire. Your fingers now naturally curl round in the
direction of the field lines.
Magnetic Induction
The strength of a magnetic field at a point is called the magnetic induction and is
denoted by the letter B. The direction of B at any point is the direction of the magnetic
field at that point.
Definition of the Tesla, the unit of magnetic induction
One tesla (T) is the magnetic induction of a magnetic field in which a conductor of
length one metre, carrying a current of one ampere perpendicular to the field is acted
on by force of one newton.
Magnitude of force on a current carrying conductor in a magnetic field
The force on a current carrying conductor depends on the magnitude of the current, the
magnetic induction and the length of wire inside the magnetic field. It also depends on
the orientation of the wire to the lines of magnetic field.
F = IlBsinθ
Where θ is the angle between the wire and magnetic field
The force is maximum when the current is perpendicular to the magnetic induction.
17
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The direction of the force on a current carrying conductor in a magnetic field
The direction of the force is perpendicular to the plane containing the wire and the
magnetic induction. When θ is 90o the force is perpendicular to both the current and
the magnetic induction.
Right hand rule: using the right hand hold the thumb and first two
fingers at right angles to each other. Point the first finger in the
direction of the field, the second finger in the direction of the
electron flow, then the thumb gives the direction of the thrust, or
force.
Note: the direction of the force will reverse if the current is
reversed.
Example
A wire, which is carrying a current of 6.0 A, has 0.50 m of its length placed in a
magnetic field of magnetic induction 0.20 T. Calculate the size of the force on the wire
if it is placed:
(a) at right angles to the direction of the field,
(b) at 45° to the to the direction of the field and,
(c) along the direction of the field (i.e. lying parallel to the field lines).
Solution
(a)
(b)
(c)
F IVB sin θ IVBsin90°
F 6.0×0.50×0.20×1
F 0.60N
F IVB sin θ IVBsin45°
F 6.0×0.5×0.20×0.707
F 0.42N
if θ = 0° sin θ = 0 F = 0 N
Magnetic induction at a distance from a long current carrying wire
The magnetic induction around an "infinitely" long current carrying conductor placed
in air can be investigated using a Hall Probe*(see footnote). It is found that the
magnetic induction B varies as I, the current in the wire, and inversely as r, the distance
from the wire.
B
μ0 I
2πr
where µo is the permeability of free space.
µo serves a purpose in magnetism very similar to that played by εo in electrostatics.
The definition of the ampere fixes the value of µo exactly.
*Footnote: A Hall Probe is a device based around a thin slice of n or p-type
semiconducting material. When the semiconducting material is placed in a magnetic
field, the charge carriers (electrons and holes) experience opposite forces which cause
them to separate and collect on opposite faces of the slice. This sets up a potential
difference - the Hall Voltage. This Hall Voltage is proportional to the magnetic
induction producing the effect.
18
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Force per unit length between two parallel wires
Two adjacent current carrying wires will influence one another due to their magnetic
fields. For wires separated by distance r , the magnetic induction at wire 2 due to the
current in wire 1 is given by:
B1 μ0 I1
2πr
Thus wire 2, carrying current I2 will experience a force:
F1→2 I2 VB1
along length l
Substitute for B1 in the above equation:
μ I
0 1
F1→2 I2 V 2πr
μ I1 I2
F
0
V
2πr
F
Z
is known as the force per unit length.
Direction of force between two current carrying wires
Wires carrying current in the same direction will attract.
Wires carrying currents in opposite directions will repel.
This effect can be shown by passing fairly large direct currents through two strips of
aluminium foil separated by a few millimetres. The strips of foil show the attraction
and repulsion more easily if suspended vertically. A car battery could be used as a
supply.
Definition of the Ampere
A current of one ampere is defined as the constant current which, if in two straight
parallel conductors of infinite length placed one metre apart in a vacuum, will produce
a force between the conductors of 2 x 10-7 newtons per metre.
[
To confirm this definition apply L Z \ ] ^
0 1
2_F
M to this situation.
Thus I1 and I2 both equal 1 A, r is 1 m and µo = 4π x 10-7 N A-2.
μ I1 I2
F
0
V
2πr
4π×10`7 ×1×1
2π×1
2 × 10`7 Nm-1 .
Equally, applying this definition fixes the value of µo = 4π x 10-7 N A-2.
We will see later that the usual unit for µo is H m-1 which is equivalent to N A-2.
19
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Comparing Gravitational, Electrostatic and Magnetic Fields
Experimental results and equations
(a) Two masses exert a force on each
other.
Force, F =
Field concept
Either mass is the source of a
gravitational field and the other mass
experiences a force due to that field.
G m1 m2
r2
At any point, the gravitational field
strength g (N kg-1) is the force acting on
a one kg mass placed at that point.
(b) Two stationary electric charges exert a
force on each other.
Force, F =
Either charge is the source of an
electrostatic field and the other charge
experiences a force due to that field.
Q1 Q2
4πε0 r 2
At any point, the electric field strength E
(N C-1) is the force acting on +1 C of
charge placed at that point.
Either current-carrying wire is the
source of a magnetic field and the other
current-carrying wire experiences a
force due to that field.
(c) Two parallel current-carrying wires
exert a force on each other.
Force for one metre of wire, F is given by
F
μ0 I1 I2
V
2πr
The force between such wires is due to the
movement of charge carriers, the current.
At any point, the magnetic induction B
is given by
μ I
0
B 2πr
.
20
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Motion in a magnetic field
Magnetic Force on Moving Charges
The force on a wire is due to the effect that the magnetic field has on the individual
charge carriers in the wire. We will now consider magnetic forces on charges which
are free to move through regions of space where magnetic fields exist.
Consider a charge q moving with a constant speed v perpendicular to a magnetic field
of magnetic induction B.
We know that F = IlBsin θ
Consider the charge q moving through a distance l. (The italic l is used to avoid
confusion with the number one or a capital i.)
Z
q
qv
Then time taken to traverse the wire t v and current I t Z giving l] qv.
Substituting into F = IlBsin θ, with sin θ = 1 since θ = 90o, gives:
F = qvB
The direction of the force is given by the same right hand rule mentioned for the force
on a current carrying conductor. You should be able to state the direction of the force
for both positive and negative charges.
Note: If the charge q is not moving perpendicular to the field then the component of
the velocity v perpendicular to the field must be used in the above equation.
Motion of Charged Particles in a Magnetic Field
The direction of the force on a charged particle in a magnetic field is perpendicular to
the plane containing the velocity v and magnetic induction B. The magnitude of the
force will vary if the angle between the velocity vector and B changes. The examples
which follow illustrate some of the possible paths of a charged particle in a magnetic
field.
Charge moving parallel or antiparallel to the magnetic field
The angle θ between the velocity vector and the magnetic field direction is 0° or 180°
hence the force F = 0. The path is a straight line.
The direction of the charged particle is not altered.
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Charge moving perpendicular to the magnetic field
If the direction of v is perpendicular to B, then θ = 90° and sin θ = 1. Now, F = qvB.
The direction of the force F is perpendicular to the plane containing v and B.
A particle travelling at constant speed under the action of a force at right angles to its
velocity will move in a circle, as it is a central force that acts on the particle. This
central force is studied in the Mechanics unit.
The sketch below shows this situation. (Remember an X indicates that the direction of
the field is 'going away' from you 'into the paper'.)
The charged particle will move in a circle, of radius r. The magnetic force supplies the
central acceleration, and maintains the circular motion.
Thus:
qvB mv 2
r
giving the radius
r mv
qB
v
The frequency of the rotation can be determined using angular velocity ω r and
ω 2πf and substituting in the above equation, giving f qB
.
2πm
Charge moving at an angle to the magnetic field
If the velocity vector v makes an angle θ with B, the particle moves in a helical
motion, the central axis of which is parallel to B.
The helix is obtained from the sum of
two motions:
• a uniform circular motion, with a
constant speed v sin θ in a plane
perpendicular to the direction of B.
• a uniform speed of magnitude v
cos θ along the direction of B
The frequency of the rotation is f qB
2πm
giving the period T 2πm
qB
, the time
between similar points. The pitch p of the helix, shown on the sketch, is the distance
between two points after one period and is given by p = v (cos θ) T.
It is worth looking into the behaviour of this equation as θ reaches either 0° or 90°.
22
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•
•
•
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Notes
The orbit frequency does not depend on the speed v or radius r. It is dependent on the
c
charge to mass ratio ( ) and the magnetic induction B.
Positive charges will orbit in the opposite direction to negative charges, since force F
is reversed.
Particles, having the same charge but different masses, e.g. electrons and protons,
entering the magnetic field along the same line will have different radii of orbit.
The kinetic energy of the particle in orbit is a constant because its orbital speed is
constant. The magnetic force does no work on the charges.
Deflections of charged particles in a bubble chamber
The diagram below is an diagram of a photograph taken in a bubble chamber in which
there is a strong magnetic field. The detecting medium is liquid hydrogen. The
ionisation associated with fast moving charged particles leaves a track of hydrogen gas
(bubbles). The magnetic field is perpendicular to the container (into or out of the page).
This allows positive and negative particles to separate and be measured more easily.
The tracks of two particles in a bubble chamber, one electron and one positron are
created by an incoming gamma ray photon, γ.
Notice the nature and directions of the deflections.
As the particles lose energy their speed decreases and the radius decreases.
Note: Problems involving calculations on the motion of charged particles in magnetic fields
should involve non-relativistic velocities only. Although in many practical applications
electrons do travel at high velocities, these situations will not likely be assessed.
23
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Applications of Electromagnetism
When electric and magnetic fields are combined in certain ways many useful devices
and measurements can be devised.
The Cyclotron
This device accelerates charged particles such as
protons and deuterons. Scientists have discovered a
great deal about the structure of matter by examining
high energy collisions of such charged particles with
atomic nuclei.
The cyclotron comprises two semi-circular D-shaped
structures (D’s).
There is a gap between the dees across which there is
an alternating voltage.
Towards the outer rim there is an exit hole through which the particle can escape;
radius = R. From this point the particle is directed towards the target.
Charged particles are generated at the ion source and allowed to enter the cyclotron.
Every time an ion crosses the gap between the D’s it gains kinetic energy, qV, due to
acceleration by the electric field. For this to happen in step, the frequency of the a.c.
must be the same as the cyclotron frequency, f.
f qB
2πm
and
r mv
qB
Thus, radius increases as velocity increases. At R, velocity will be at its maximum:
d
ef cgh
and
1
Ek on exit = mv 2 2
q2 B2 R2
2m
The Velocity Selector and Mass Spectrometer
Charged particles can be admitted to
a region of space where electric and
magnetic fields are 'crossed', i.e.
mutually perpendicular. Particles
can only exit via a small slit as shown
below.
Magnetic field is uniform and is
directed 'into the paper'.
Electric field
Electric deflecting force
V
E= d
Fe = qE
Beyond the exit slit, the particles only experience a magnetic field.
Magnetic force, Fm = qvB and its direction is as shown.
E
If particle is undeflected, Fm = Fe (in magnitude) thus qvB = qE and v B.
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Hence only charges with this specific velocity will be selected. Note that this
expression is independent of q and m. Thus this device will select all charged particles
which have this velocity.
In the mass spectrometer ions are selected which have the same speed. After leaving
the velocity selector they are deflected by the magnetic field and will move in a circle
i
, as shown previously. The ions tend to have lost one electron so
of radius, F
cg
have the same charge. Since their speed is the same the radius of path of a particular
ion will depend on its mass. Thus the ions can be identified by their deflection.
JJ Thomson's Experiment to Measure the Charge to Mass Ratio of Electrons
This method uses the crossed electric and magnetic fields mentioned above.
The electric field E is applied by the p.d. Vp across the plates. The separation of the
plates is d and their length is L. The current in the Helmholtz coils is slowly increased
until the opposite magnetic deflection cancels out the electric deflection and the
electron beam appears undeflected. The value of current, I, at this point is noted.
Using the magnetic field only, the deflection, y, of the beam is recorded.
For the undeflected beam: Fmag Felec
qvB qE
E
v ----------- (1)
B
For the magnetic field only, the central force is provided by the magnetic field
q e vB qe
m
mv 2
where r is the radius of curvature
r
v
rB
------------ (2)
Eliminate v between equations (1) and (2).
qe
m
E
Vp
rB2 rB2 d
using E
Vp
d
The plate separation d and Vp are easily measured, r is determined from the deflection
y using r = (L2+y2)/2y. B is found using the current I and the Helmholtz coil relation,
8k NI
(B where N is the number of turns and r the radius of the coils).
√125r
25
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Capacitors
Capacitors in d.c. circuits
Consider the following circuit:
When the switch is set to position B the capacitor will charge. When the switch is set
to position A the capacitor discharges.
The current through the capacitor and the voltage across it can be monitored to obtain
graphs showing values over time.
Charging
Discharging
Although the discharging current/time graph has the same shape as that during
charging, the currents in each case are flowing in opposite directions. The discharging
current decreases because the pd across the plates decreases as charge leaves them.
A capacitor stores charge, but unlike a cell it has no capability to supply more energy.
When it discharges, the energy stored will be used in the circuit, e.g. in the above circuit
it would be dissipated as heat in the resistor.
26
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Factors affecting the rate of charge and discharge
The time taken for a capacitor to charge is controlled by the resistance of the resistor,
R, (because it controls the magnitude of the current, i.e. the charge flow rate) and the
capacitance of the capacitor, C, (since a larger capacitor will take longer to fill with
charge or to empty). As an analogy, consider charging a capacitor as being like filling
a jug with water. The size of the jug is like the capacitance and the resistor is like the
tap you use to control the rate of flow.
The values of R and C can be multiplied together to form what is known as the time
constant. Can you prove that R × C has units of time, seconds? The time taken for the
capacitor to charge or discharge is related to the time constant.
Large capacitance and large resistance both increase the charge or discharge time.
The current/time graphs for capacitors of different value during charging are shown
below:
The effect of capacitance
The effect of resistance
Note that since the area under the current/time graph is equal to charge, for a given
capacitor the area under the graphs must be equal.
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Time constant
The time it takes a capacitor to discharge through a resistor depends on capacitance, C,
of the capacitor and the resistance, R, of the resistor.
When a capacitor, C, is charged to a p.d. V0 it stores a charge, Q0, since Q0 = CV0.
When the capacitor is discharged through resistor, R, the current I = V/R, where V is
the p.d. across C.
dQ
The current at time, t, during the discharge is also given by I - . The negative sign
dt
indicates that Q decreases with time.
Since
V
I
R
dQ
Q
- dt CR
Q dQ
mQ
0
Q
-
t
1
m dt
CR 0
1
nlnQoQQ0 - ntot0
CR
Q
t
ln pQ q - CR
0
Q Q0 e$-t⁄CR'
Hence charge, Q, decreases exponentially with time, t. Since the potential difference.
V, across C is directly proportional to Q it follows that V V est/vw .
In addition, since the current, I, in the circuit is directly proportional to V, then
I I est/vw where I0 is the initial current value and I0 = V0/R.
From Q
by
Q est/vw , Q decreases from Q0 to half its value, Q0/2, in a time, t1 given
est/vw @ 2s
t
CRln2
Similarly Q decreases from Q0/2 to Q0/4 in time t1. Thus the time for a charge to
decrease from any value to half of that value is always the same.
The time constant, T, of the discharge circuit is defined as CR seconds, where C is the
capacitance in farads and R is the resistance in ohms.
Hence if t = T = CR then
1
Q Q0 e-1 e Q0
Therefore the time constant can be defined as
the time for the charge to decay to 1/e times its
initial value. Since e = 2.72. 1/e = 0.37. If the
time constant is high, then the charge will decay
slowly, if the time constant is small, then the
charge will decay rapidly.
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An uncharged capacitor, C, is charged through a resistor, R, by a battery of emf, E, and
negligible internal resistance. Initially the capacitor has no charge stored and hence no
p.d. across it. Therefore the initial current I0 = E/R. If I is the current flowing after
time, t, and the p.d. across the capacitor is VC then
I
E-VC
R
But I = dQ/dt and VC = Q/C so:
Q
E-p xCq
dQ
dt
R
dQ
CR dt CE-Q Q0 -Q
where Q0 = CE = final charge on C when no current flows.
Integrating
1
t
Q dQ
t
-$lnnQ 0 -Qo-lnQ 0 '
t
-ln p
m dt
CR 0
CR
CR
e-t⁄CR m0
Q 0 -Q
Q 0 -Q
Q0
q
Q 0 -Q
Q0
Q Q 0 R1-e-t⁄CR S
Where the time constant CR is large then it takes a long time for the capacitor to reach
its final charge, that is the capacitor charges slowly. If the time constant is small the
capacitor charges quickly. The p.d. across the capacitor, VC, shows the same variation
as Q since VC ∝ Q.
29
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Capacitors in a.c. circuits
Consider the following circuit:
The signal generator is set to a low frequency and the potential difference across the
resistor to a known voltage. The current in the circuit is then measured using the
ammeter.
The frequency of the signal generator is altered but the potential difference kept
constant and the current measured.
The current in the resistor remains constant with frequency with I = V/R. Resistors are
unaffected by the frequency of the supply and behave in the same way in both d.c. and
a.c. circuits.
The resistor is replaced with a capacitor and the experiment repeated.
The current in the circuit increases in direct proportion to the frequency.
Capacitive Reactance
The opposition to current in a capacitive circuit is known as the capacitive reactance,
XC.
Current in a capacitive circuit is determined by I = V/XC and therefore XC = V/I. XC
is measured in ohms, the same as resistance, but it is not appropriate to refer to the
opposition to current in a capacitive circuit as a resistance.
I ∝ f and XC ∝ 1/I therefore XC ∝ 1/f.
XC 1
2πfC
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Inductors
Electromagnetic Induction
Our present day large scale production and distribution of electrical energy would not
be economically feasible if the only source of electricity we had came from chemical
sources such as dry cells. The development of electrical engineering began with the
work of Michael Faraday and Joseph Henry. Electromagnetic induction involves the
transformation of mechanical energy into electrical energy.
A Simple Experiment on Electromagnetic Induction
Apparatus: coil, magnet, centre-zero meter
Observations
1. When the magnet is moving into coil - meter
needle moves to the right (say). We say a current
has been induced.
2. When the magnet is moving out of the coil induced current is in the opposite direction
(left).
3. Magnet stationary, either inside or outside the coil - no induced current.
4. Moving the magnet faster makes the induced current larger.
5. When the magnet is reversed, i.e. the south pole is nearest the coil, - induced current
reversed.
Note: Moving the coil instead of the magnet produces the same effect. It is the
relative movement which is important.
The induced currents that are observed are said to be produced by an induced
electromotive force, e.m.f. This electrical energy must come from somewhere. The
work done by the person pushing the magnet at the coil is the source of the energy. In
fact the induced current sets up a magnetic field in the coil which opposes the movement
of the magnet.
Summary
The size of the induced e.m.f. depends on:
• the relative speed of movement of the magnet and coil
• the strength of the magnet
• the number of turns on the coil.
Self inductance
A current in a coil sets up a magnetic field through and round the coil. When the current
in the coil changes the magnetic field changes. A changing magnetic field induces an
e.m.f across the coil. This is called a self induced e.m.f. because the coil is inducing an
e.m.f. in itself due to its own changing current.
The coil, or inductor as it is called, is said to have the property of inductance, L.
The inductance of an inductor depends on its design. Inductance is a property of the
device itself, like resistance of a resistor or capacitance of a capacitor. An inductor will
tend to have a large inductance if it has many turns of wire, a large area and is wound
on an iron core.
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Growth and Decay of Current in an Inductive Circuit.
An inductor is a coil of wire wound on a soft iron
core.
An inductor is denoted by the letter L.
An inductor has inductance, see later.
Growth of current
Switch S2 is left open.
When switch S1 is closed the ammeter reading rises slowly to a final value, showing
that the current takes time to reach its maximum steady value. With no current there is
no magnetic field through the coil. When S1 is closed the magnetic field through the
coil will increase and an e.m.f. will be generated to prevent this increase. The graph
below shows how the circuit current varies with time for inductors of large and small
inductance.
Decay of current
After the current has reached its steady value S2 is closed, and then S1 is opened. The
ammeter reading falls slowly to zero. The current does not decay immediately
because there is an e.m.f. generated which tries to maintain the current, that is the
induced current opposes the change. The graph below shows how the circuit current
varies with time for inductors of large and small inductance.
Notes
• For both the growth and decay, the induced e.m.f. opposes the change in current.
• For the growth of current, the current tries to increase but the induced e.m.f. acts to
prevent the increase. It takes time for the current to reach its maximum value.
Notice that the induced e.m.f. acts in the opposite direction to the circuit current.
• For the decay of current the induced e.m.f acts in the same direction as the current
in the circuit. Now the induced e.m.f. is trying to prevent the decrease in the current.
32
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Experiment to show build up current in an inductive circuit
The switch is closed and the variable resistor
adjusted until the lamps B1 and B2 have the
same brightness.
The supply is switched off.
The supply is switched on again and the
brightness of the lamps observed.
Lamp B2 lights up immediately. There is a time lag before lamp B1 reaches its
maximum brightness. An e.m.f. is induced in the coil because the current in the coil is
changing. This induced e.m.f. opposes the change in current, and it is called a back
e.m.f. It acts against the increase in current, hence the time lag.
The experiment is repeated with an inductor of more turns. Lamp B1 takes longer to
light fully. If the core is removed from the inductor, lamp B1 will light more quickly.
Induced e.m.f. when the current in a circuit is switched off
When the current in a circuit, containing an inductor, is switched off the magnetic field
through the inductor will collapse very rapidly. There will be a large change in the
magnetic field leading to a large induced e.m.f. For example a car ignition coil produces
a high e.m.f. for a short time when the circuit is broken.
Lighting a neon lamp
The 1.5 V supply in the circuit below is insufficient to light the neon lamp. A neon
lamp needs about 80 V across it before it will light.
The switch is closed and the current builds up to
its maximum value. When the switch is opened,
the current rapidly falls to zero. The magnetic
field through the inductor collapses (changes) to
zero producing a very large induced e.m.f. for a
short time. The lamp will flash.
Circuit symbols for an inductor
An inductor is a coil of wire which may be wound on a magnetic core, e.g. soft iron,
or it may be air cored.
Inductor with a core
Inductor without a core
Conservation of Energy and Direction of Induced e.m.f.
In terms of energy, the direction of the induced e.m.f. must oppose the change in
current. If it acted in the same direction as the increasing current we would be able to
produce more current for no energy! This would violate the conservation of energy.
The source has to do work to drive the current through the coil. It is this work done
which appears as energy in the magnetic field of the coil and can be obtained when the
magnetic field collapses, e.g. the large e.m.f. generated for a short time across the neon
lamp.
33
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Lenz's Law.
Lenz’s laws summarises this. The induced e.m.f. always acts in such a direction as
to oppose the change which produced it. Anything which causes the magnetic field
in a coil to change will be opposed.
An inductor is sometimes called a ‘choke’ because of its opposing effects.
However it must be remembered that when the current decreases the effect of an
inductor is to try and maintain the current. Now the induced e.m.f. acts in the same
direction as the current, yet still against the change.
Magnetic flux - an aside to clarify terminology
Magnetic flux may be thought of as the number of lines of a magnetic field which pass
through a coil.
Angle is also an important factor. At 90° the flux drops to zero, since there are no
lines intersecting the loop.
Faradays laws refer to the magnetic flux ϕ rather than the magnetic induction B. His
two laws are given below.
1. When the magnetic flux through a circuit is changing an e.m.f. is induced.
2. The magnitude of the induced e.m.f. is proportional to the rate of change of the
magnetic flux.
Magnitude of the induced e.m.f.
The self-induced e.m.f. E in a coil when the current I changes is given by
E -L
dI
dt
where L is the inductance of the coil.
The negative sign indicates that the direction of the e.m.f. is opposite to the change in
current.
The inductance of an inductor can be determined experimentally by measuring the
dI
e.m.f. and rate of change of current, . This is usually done by finding the gradient of
dt
start of the growth curve on the current/time graph for an inductor, i.e. when the back
e.m.f. is equal and opposite to the circuit e.m.f. and the cicuit current is zero.
34
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Definition of Inductance
The inductance, L, of an inductor is one henry (H) when an e.m.f. of one volt is induced
across the ends of the inductor when the current in the inductor changes at a rate of one
ampere per second.
A comment on units
The unit for permeability µo was stated to be N A-2 with a usual unit of H m-1. From
the above formula, in terms of units, we can see that the e.m.f (joules per coulomb)
J C-1 = H A s-1 which is N m (A s)-1 = H A s-1
giving N m A-1 s-1 = H A s-1 and N A-2 = H m-1 (the s-1 cancels)
Energy stored by an Inductor
In situations where the current in an inductor is suddenly switched off large e.m.f.s are
produced and can cause sparks. At the moment of switch off the change in current is
very large. The inductor tries to maintain the current as the magnetic field collapses
and the energy stored by the magnetic field is given up. A magnetic field can be a
source of energy. To have set up the magnetic field work must have been done.
Equation for the energy stored in an inductor
For an inductor with a current I the energy stored is given by the equation below:
1
E LI2
2
where L is the inductance of the inductor and I the steady current.
Example
An inductor is connected to a 6.0 V d.c. supply which has a negligible internal
resistance. The inductor has a resistance of 0.8 Ω. When the circuit is switched on it
is observed that the current increases gradually. The rate of growth of the current is
200 A s-1 when the current in the circuit is 4.0 A.
Here a resistor is used to represent
the resistance of the inductor.
(a)
(b)
(c)
(d)
Calculate the induced e.m.f. across the coil when the current is 4.0 A.
Hence calculate the inductance of the coil.
Calculate the energy stored in the inductor when the current is 4.0 A.
(i) When is the energy stored by the inductor a maximum?
(ii)What value does the current have at this time?
Solution
(a) Potential difference across the resistive element of the circuit V = I R
4 x 0.8 = 3.2 V
Thus p.d. across the inductor = 6.0 - 3.2 = 2.8 V
dI
2.8
(b)
Using E -L dt gives L 200 0.014H 14mH
1
(c)
Using E 2 LI2 0.5 0.014 4 4 0.11J
(d) (i) The energy will be a maximum when the current reaches a steady value.
emf
6.0
(ii)Imax R 0.8 7.5A
35
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Inductors in a.c. circuits
In an a.c. circuit the current is continually changing. This means that the magnetic field
through the inductor is continually changing. Hence an e.m.f. is continually induced in
the coil.
Consider the applied alternating voltage at the point in the cycle when the voltage is
zero. As the current tries to increase the induced e.m.f. will oppose this increase. Later
in the cycle as the voltage decreases the current will try to fall but the induced e.m.f.
will oppose the fall. The induced e.m.f. produced by the inductor will continually
oppose the current.
If the frequency of the applied voltage is increased then the rate of change of current
increases. The magnitude of the induced e.m.f. will also increase. Hence there should
be a greater opposition to the current at a higher frequency.
Frequency response of inductor
An inductor is connected in series with
an alternating supply of variable
frequency and constant amplitude.
Readings of current and frequency are
taken.
As the frequency is increased the current is observed to decrease. The opposition to
the current is greater at the higher frequencies. Graphs of current against frequency
and current against 1/frequency are shown below.
Note: these graphs show the inductive effects only. Considering the construction of
an inductor, it is likely that the inductor has some resistance. A 2400 turns coil has a
resistance of about 80 Ω. The opposition to the current at ‘zero’ frequency will be the
resistance of the inductor. In practice if readings were taken at low frequencies, the
current measured would be a mixture of the inductive and resistive effects.
An inductor can be used to block a.c. signals while transmitting d.c. signals, because
the inductor produces large induced e.m.f.s at high frequencies.
For a capacitor in an a.c. circuit the current increases when the frequency increases.
The inductor has the opposite effect to a capacitor.
36
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Inductive Reactance
The opposition to current in an inductive circuit is known as the inductive reactance,
XL.
Current in an inductive circuit is determined by I = V/XL and therefore XL = V/I. XL is
measured in ohms, the same as resistance, but it is not appropriate to refer to the
opposition to current in a capacitive circuit as a resistance.
I ∝ 1/f and XL ∝ 1/I therefore XL ∝ f.
XL = 2πfC
Combining Inductive and Capacitive Reactance and Resistance
In inductive and capacitive circuits reactances and resistance cannot be added
arithmetically. There are phase differences between inductive and capacitive
reactances and resistances and therefore vector addition must be used. The total
impedance Z, measured in ohms, in an a.c. circuit is given by:
}
~$•€ ` •• '@ 3 ‚ @
Uses of inductors and capacitors in a.c. circuits
The circuit below shows a capacitor and inductor in series with an alternating supply.
At low frequencies the opposition to the current by the inductor is low, so the p.d across
L will be low. At low frequencies the opposition to the current by the capacitor is high
so the p.d. across C will be high. At low frequencies XL<XC.
At high frequencies, the reverse is the case. The p.d. across the inductor VL will be the
higher than the p.d. across the capacitor. At high frequencies XL>XC.
This circuit could be used to filter high and low frequency signals.
37
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Cross-over Networks in Loudspeakers
Capacitor C1 allows high frequency signals to
pass to loudspeaker S1.
High frequency signals can also pass more
easily through capacitor C2 than loudspeaker
S2.
Low frequency signals are ‘blocked’ by C1
and C2 but pass easily through inductor L to
loudspeaker S2.
Capacitors in Radio Circuits
Capacitor C1 is a variable capacitor, which when used in conjunction with inductor L,
allows the radio to be tuned to one particular radio frequency. Capacitor C2 allows the
high frequency radio carrier signal to flow to earth but ‘blocks’ the low frequency audio
signal which must then pass on to the amplifier and loudspeaker of the radio.
38
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
Amplifier Bias Network
When an a.c. signal is to be amplified by a
simple transistor amplifier the a.c. signal
should be input to the transistor via a
capacitor.
The capacitor will allow the a.c. signal to
pass but ‘block’ any unwanted d.c. signal.
The transformer
The principle of operation of the transformer can be given in terms of induced e.m.f.
When S is closed, the meter needle
‘kicks’ momentarily then returns to
zero. When the current is steady the
meter reads zero. When S is opened,
the meter needle kicks briefly in the
opposite direction and returns to zero.
A changing magnetic field is produced
in the coil when the current changes.
This changing magnetic field will
produce an induced e.m.f. during this
short time. However when the current
is steady, there is no changing field
hence no induced e.m.f.
With an a.c. supply the current is
continually changing. This sets up a
continually changing magnetic field in
the soft iron core. Hence an induced
e.m.f. is produced in the secondary coil.
From the conservation of energy the
direction of the induced e.m.f. will
oppose the change which sets it up.
Hence the direction of a current in the
secondary, at any time, will always be in
the opposite direction to the current in
the primary.
We can now understand why a transformer only operates with an alternating supply.
The transformer is an example of mutual inductance.
39
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council
Curriculum For Excellence
Advanced Higher Physics
Electromagnetism
The unification of electricity and magnetism
In the 1860s James Clerk Maxwell unified electricity and magnetism using four
equations. One of the outcomes of these equations was the prediction of
electromagnetic waves.
Electromagnetic waves have both electric and magnetic field components which
oscillate in phase, perpendicular to each other and to the direction of energy
propagation.
The diagram below shows a 3-dimensional picture of such a wave.
The above diagram shows the variation of the electric field strength, E, in the x-z
plane and the variation of the magnetic induction, B, in the x-y plane.
Maxwell’s equations result in the relationship between the speed of light and the
permittivity and permeability of free space.
c 1
~μ0 ε0
This means that all electromagnetic waves, regardless of frequency or wavelength,
travel at a constant speed in a vacuum.
c 1
√4π 10-7 8ƒ85 10-12
c = 3·00 × 108 m s−1
40
Compiled and edited by F. Kastelein
Source: RGC, LTS
Boroughmuir High School
City of Edinburgh Council